Related papers: The mean king's problem: Prime degrees of freedom
Given a separable nonconstant polynomial $f(x)$ with integer coefficients, we consider the set $S$ consisting of the squarefree parts of all the rational values of $f(x)$, and study its behavior modulo primes. Fixing a prime $p$, we…
We consider the problem of determining the state of a quantum system given one or more readings of the expectation value of an observable. The system is assumed to be a finite dimensional quantum control system for which we can influence…
Every natural number greater than two may be written as the sum of a prime and a square-free number. We establish several generalisations of this, by placing divisibility conditions on the square-free number.
We prove a generalization of the author's work to show that any subset of the primes which is `well-distributed' in arithmetic progressions contains many primes which are close together. Moreover, our bounds hold with some uniformity in the…
We show that every sufficiently large integer is a sum of a prime and two almost prime squares, and also a sum of a smooth number and two almost prime squares. The number of such representations is of the expected order of magnitude. We…
We prove that every integer greater than two may be written as the sum of a prime and a square-free number.
We present the technique of derivation of a theory to obtain an $(n+1)f$-degrees-of-freedom theory from an $f$-degrees-of-freedom theory and show that one can calculate all of the quantities of the derived theory from those of the original…
Free probability analogs of the basics of extreme-value theory are obtained, based on Ando's spectral order. This includes classification of freely max-stable laws and their domains of attraction, using ``free extremal convolutions'' on the…
Joint degree vectors give the number of edges between vertices of degree $i$ and degree $j$ for $1\le i\le j\le n-1$ in an $n$-vertex graph. We find lower and upper bounds for the maximum number of nonzero elements in a joint degree vector…
We investigate average gradient degree of normal random polynomials of fixed algebraic degree n. In particular, for polynomials of two variables, asymptotics of the average gradient degree for large values of n is determined.
Let f be an irreducible polynomial of degree d>=3 with no fixed prime divisor. We derive an asymptotic formula for the number of primes p<x such that f(p) is (d-1)-free.
We calculate admissible values of r such that a square-free polynomial with integer coefficients, no fixed prime divisor and irreducible factors of degree at most 3 takes infinitely many values that are a product of at most r distinct…
Recently a class of generalized information measures was defined on sets of items parametrized by submodular functions. In this paper, we propose and study various notions of independence between sets with respect to such information…
Using evaluations of the difference between consecutive primes we develop another way of estimating of the number of primes in the interval $(n, 2n)$. We also discuss the ultra Cramer conjecture, $p_{n+1} - p_n = O(log^{1+\epsilon}p_n)$…
We estimate the number of primes represented by a general quadratic polynomial with discriminant $\Delta$, assuming that the corresponding real character is exceptional.
We consider the representation of primes as a sum of a prime and twice a triangular number. We prove that a subset of the primes having density 1 is expressible in this form. We conjecture that every odd prime number is expressible as a sum…
This is the first of two coupled papers estimating the mean values of multiplicative functions, of unknown support, on arithmetic progressions with large differences. Applications are made to the study of primes in arithmetic progression…
Specification of the strongest possible Bell inequalities for arbitrarily complicated physical scenarios -- any number of observers choosing between any number of observables with any number of possible outcomes -- is currently an open…
For $k\ge1$, a $k$-almost prime is a positive integer with exactly $k$ prime factors, counted with multiplicity. In this article we give elementary proofs of precise asymptotics for the reciprocal sum of $k$-almost primes. Our results match…
We study correlations in a bipartite, Fermionic, free state in terms of perturbations induced by one party on the other. In particular, we show that all so conditioned free states can be modelled by an auxiliary Fermionic system and a…